Explanation Given: The equation of the first octant of the ball is x 2 + y 2 + z 2 = 1 and the mass density is ρ ( x , y , z ) = x Formula Used: The center of mass can be calculated as: ( x C M , y C M , z C M ) = ( 1 M ∫ ∫ W ∫ x ρ ( x , y , z ) d V , 1 M ∫ ∫ W ∫ y ρ ( x , y , z ) d V , 1 M ∫ ∫ W ∫ z ρ ( x , y , z ) d V ) M = ∫ ∫ W ∫ ρ ( x , y , z ) d V Calculation: The spherical coordinates are x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ , z = ρ cos ϕ 0 ≤ ρ ≤ 1 , 0 ≤ θ ≤ π 2 , 0 ≤ ϕ ≤ π 2 d V = ρ 2 sin ϕ d ρ d θ d ϕ Evaluating M = ∫ ∫ W ∫ ρ ( x , y , z ) d V , ∫ ∫ W ∫ ρ ( x , y , z ) d V = ∫ ϕ = 0 π 2 ∫ θ = 0 π 2 ∫ ρ = 0 1 ( ρ sin ϕ cos θ ) ρ 2 sin ϕ d ρ d ϕ d θ = [ ∫ ρ = 0 1 ρ 3 d ρ ] [ ∫ ϕ = 0 π 2 sin 2 ϕ d ϕ ] [ ∫ θ = 0 π 2 cos θ d θ ] = [ ρ 4 4 ] 0 1 [ sin θ ] 0 π 2 [ 1 2 × π 2 ] = [ 1 4 ] [ 1 ] [ π 4 ] = π 16 Evaluating ∫ ∫ W ∫ x ρ ( x , y , z ) d V , ∫ ∫ W ∫ x ρ ( x , y , z ) d V = ∫ ϕ = 0 π 2 ∫ θ = 0 π 2 ∫ ρ = 0 1 ( ρ sin ϕ cos θ ) ρ 2 sin ϕ d ρ d ϕ d θ = ∫ ϕ = 0 π 2 ∫ θ = 0 π 2 ∫ ρ = 0 1 ρ 4 sin 3 ϕ cos 2 θ d ρ d ϕ d θ = [ ∫ ρ = 0 1 ρ 4 d ρ ] [ ∫ θ = 0 π 2 cos 2 θ d θ ] [ ∫ ϕ = 0 π 2 sin 3 ϕ d ϕ ] = [ ρ 5 5 ] 0 1 [ 1 2 π 2 ] [ 2 3 × 1 ] = [ 1 5 ]

4th Edition

ISBN: 9781319434540

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16, Problem 51CRE

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To calculate:

The center of mass of the first octant of the ball $x^{2} + y^{2} + z^{2} = 1$

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Chapter 16, Problem 50CRE

Chapter 16, Problem 52CRE

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Chapter 16 Solutions

ACHIEVE STANDALONE ACCESS F/ CALC 4E

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The center of mass of the first octant of the ball x 2 + y 2 + z 2 = 1

Solution Summary: The author explains how to calculate the center of mass of the first octant of a ball. The spherical coordinates are x, y, and z.

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To calculate:

The center of mass of the first octant of the ball $x^{2} + y^{2} + z^{2} = 1$

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Chapter 16 Solutions

The center of mass of the first octant of the ball x 2 + y 2 + z 2 = 1

Solution Summary: The author explains how to calculate the center of mass of the first octant of a ball. The spherical coordinates are x, y, and z.

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To calculate: The center of mass of the first octant of the ball x2+y2+z2=1

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Chapter 16 Solutions

To calculate:

The center of mass of the first octant of the ball $x^{2} + y^{2} + z^{2} = 1$